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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}$I know a little bit about complex representation theory of finite reductive groups as $\GL_n(q),\SO_n(q)$ etc via Deligne-Lusztig induction and so on. If I correctly understood, there's another geometric way to build the characters (at least in the $\GL_n$ case) via the so-called character sheaves. I've just have read very vaguely something about this and it seems to me strictly related to the same circle of idea which brings up Springer correspondence etc: I'd be particularly interested in the link between the twos.

Are there any book or introductory references which treat this construction? I'd be particularly interested just in the $\GL_n$ case and in a focus towards examples maybe. I tried to read the original articles by Lusztig but they are maybe a bit too general/ difficult for what I had in mind.

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I am not exactly sure what you have read already, but how about this set of notes, from a course given by Victor Ostrik in Luminy in 2010 (notes by Geordie Williamson)?

It says in $\S~2.1$ that, given a reductive algebraic group $G/\mathbb{F}_q$, the goal is to construct a class $\widehat{G}$ of irreducible perverse sheaves on $G$ such that their characteristic functions give characters of $G(\mathbb{F}_q)$, which as far as I understand is what you are asking about :-)

If you are interested in the relations with the Springer correspondence, you can consult:

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